Logistic Regression

Under the Hood

(with as little maths as possible)

Journal articles we are using


Primary paper:

  • Stephenson, J., Bunce, C., Doré, C. J., et. al. Ophthalmic Statistics Note 11: Logistic Regression. Br J Ophthalmol 100, 1594-1595 (2016).

Uses data from:

  • Wakely, L., Rahman, R. & Stephenson, J. A comparison of several methods of macular hole measurement using optical coherence tomography, and their value in predicting anatomical and visual outcomes. Br J Ophthalmol 96, 1003-1007 (2012).

Concepts you should understand


These all play a role in logistic regression, but they are nothing more than versions of one another:

A few basic formulae

Research question


Is macular hole thickness a predictor of surgical repair success.


Regression equation for the logistic regression model was:


\[ \text{logit} \; p = 10.89 - 0.016 \; \text{x} \;\text{macular hole thickness}\;(\mu m) \]

Simulate some data


Simulate data based on this regression equation. Use:

n = 1000.

Macular hole thickness: mean = 486, sd = 142.


n <- 1000                   
set.seed(1234)
mac_hole_thickness  <-  rnorm(n, 486, 142)              # generate macular hole inner opening data with mean 486 and sd = 152
logodds_success  <-  10.89 - 0.016 * mac_hole_thickness # generate variable that is linear combination of intercept = 10.89 and coefficient for macular hole -0.016 (logit scale)
odds_success  <-  exp(logodds_success)                  # exponentiate logodds to get odds
prob_success  <-  odds_success/(1 + odds_success)       # generate probabilities from this
y_success  <-  rbinom(n, 1, prob_success)               # generate outcome variable as a function of those probabilities
df <-  data.frame(cbind(mac_hole_thickness, 
                        logodds_success, 
                        prob_success, 
                        y_success))
df <- df |> 
  filter(mac_hole_thickness > 100)                      # only include those with thickness > 100

The data

mac_hole_thickness logodds_success prob_success y_success
314.6 5.86 0.997 1
525.4 2.48 0.923 1
640.0 0.65 0.657 1
152.9 8.44 1.000 1
546.9 2.14 0.895 1
557.9 1.96 0.877 1
404.4 4.42 0.988 1
408.4 4.36 0.987 1
405.8 4.40 0.988 1
359.6 5.14 0.994 1
418.2 4.20 0.985 1
344.2 5.38 0.995 1
375.8 4.88 0.992 1
495.2 2.97 0.951 1
622.2 0.93 0.718 1
470.3 3.36 0.967 1
413.4 4.28 0.986 1
356.6 5.18 0.994 1
367.1 5.02 0.993 1
829.0 -2.37 0.085 0
505.0 2.81 0.943 1
416.3 4.23 0.986 1
423.4 4.11 0.984 1
551.3 2.07 0.888 1
387.5 4.69 0.991 1
280.4 6.40 0.998 1
567.6 1.81 0.859 1
340.6 5.44 0.996 1
483.9 3.15 0.959 1
353.1 5.24 0.995 1
642.5 0.61 0.648 1
418.5 4.19 0.985 1
385.3 4.73 0.991 1
414.8 4.25 0.986 1
254.7 6.82 0.999 1
320.2 5.77 0.997 1
176.4 8.07 1.000 1
295.6 6.16 0.998 1
444.2 3.78 0.978 1
419.8 4.17 0.985 1
691.8 -0.18 0.455 0
334.3 5.54 0.996 1
364.5 5.06 0.994 1
446.2 3.75 0.977 0
344.8 5.37 0.995 1
348.5 5.31 0.995 1
328.8 5.63 0.996 1
308.2 5.96 0.997 1
411.6 4.30 0.987 1
415.4 4.24 0.986 1
229.5 7.22 0.999 1
403.3 4.44 0.988 1
328.5 5.63 0.996 1
341.9 5.42 0.996 1
463.0 3.48 0.970 1
566.0 1.83 0.862 1
720.0 -0.63 0.348 0
376.2 4.87 0.992 1
714.0 -0.53 0.369 0
321.6 5.74 0.997 1
579.2 1.62 0.835 0
848.0 -2.68 0.064 0
481.1 3.19 0.961 1
390.9 4.64 0.990 1
484.9 3.13 0.958 1
738.3 -0.92 0.284 1
324.3 5.70 0.997 1
680.2 0.01 0.502 1
674.8 0.09 0.523 1
533.8 2.35 0.913 1
487.0 3.10 0.957 1
421.3 4.15 0.984 1
434.0 3.95 0.981 1
578.1 1.64 0.838 1
780.0 -1.59 0.169 0
464.2 3.46 0.970 1
288.5 6.27 0.998 1
383.3 4.76 0.991 1
522.7 2.53 0.926 1
441.0 3.83 0.979 1
460.8 3.52 0.971 1
461.9 3.50 0.971 1
291.1 6.23 0.998 1
461.3 3.51 0.971 1
606.7 1.18 0.765 1
585.1 1.53 0.822 1
564.1 1.86 0.866 1
428.8 4.03 0.983 1
458.8 3.55 0.972 0
316.4 5.83 0.997 1
478.5 3.23 0.962 1
522.2 2.53 0.927 1
728.2 -0.76 0.318 1
628.2 0.84 0.698 1
415.6 4.24 0.986 1
536.5 2.31 0.909 1
324.9 5.69 0.997 1
610.7 1.12 0.754 0
624.2 0.90 0.712 1
787.2 -1.71 0.154 0
544.9 2.17 0.898 1
418.6 4.19 0.985 1
495.4 2.96 0.951 1
414.6 4.26 0.986 1
368.7 4.99 0.993 1
509.7 2.73 0.939 1
358.7 5.15 0.994 1
509.9 2.73 0.939 1
536.4 2.31 0.909 0
478.6 3.23 0.962 1
458.2 3.56 0.972 1
393.8 4.59 0.990 1
328.4 5.64 0.996 1
606.6 1.18 0.766 1
489.2 3.06 0.955 1
604.0 1.23 0.773 0
309.3 5.94 0.997 1
510.0 2.73 0.939 1
581.6 1.58 0.830 1
482.3 3.17 0.960 1
458.8 3.55 0.972 1
375.0 4.89 0.993 1
778.3 -1.56 0.173 0
592.6 1.41 0.804 1
745.0 -1.03 0.263 0
497.4 2.93 0.949 1
396.3 4.55 0.990 1
271.1 6.55 0.999 1
395.7 4.56 0.990 1
518.1 2.60 0.931 1
629.9 0.81 0.692 1
521.9 2.54 0.927 1
319.6 5.78 0.997 1
581.0 1.59 0.831 1
251.7 6.86 0.999 1
434.0 3.95 0.981 1
441.1 3.83 0.979 1
209.3 7.54 0.999 1
616.6 1.02 0.736 1
397.6 4.53 0.989 1
438.6 3.87 0.980 1
684.1 -0.06 0.486 1
576.4 1.67 0.841 1
470.6 3.36 0.966 1
559.0 1.95 0.875 1
542.7 2.21 0.901 1
722.1 -0.66 0.340 0
525.2 2.49 0.923 1
557.9 1.96 0.877 1
535.4 2.32 0.911 1
432.4 3.97 0.981 1
499.9 2.89 0.947 1
718.7 -0.61 0.352 0
361.7 5.10 0.994 1
503.3 2.84 0.945 1
679.4 0.02 0.505 0
452.7 3.65 0.975 1
336.4 5.51 0.996 1
362.5 5.09 0.994 1
430.6 4.00 0.982 1
365.7 5.04 0.994 1
449.0 3.71 0.976 1
427.2 4.06 0.983 1
460.0 3.53 0.972 1
543.8 2.19 0.899 1
574.7 1.69 0.845 1
724.3 -0.70 0.332 0
476.2 3.27 0.963 1
440.4 3.84 0.979 1
694.9 -0.23 0.443 1
728.0 -0.76 0.319 0
492.1 3.02 0.953 1
438.8 3.87 0.980 1
227.2 7.25 0.999 1
686.4 -0.09 0.477 0
367.1 5.02 0.993 1
326.4 5.67 0.997 1
918.2 -3.80 0.022 0
519.4 2.58 0.930 1
481.3 3.19 0.960 1
471.8 3.34 0.966 1
624.6 0.90 0.710 0
544.8 2.17 0.898 1
615.5 1.04 0.739 1
767.7 -1.39 0.199 1
652.0 0.46 0.612 0
413.8 4.27 0.986 1
586.0 1.51 0.820 1
457.8 3.56 0.972 1
409.6 4.34 0.987 1
373.9 4.91 0.993 1
555.3 2.01 0.881 1
793.9 -1.81 0.140 0
557.1 1.98 0.878 1
574.1 1.70 0.846 1
348.8 5.31 0.995 1
509.1 2.74 0.940 1
190.9 7.84 1.000 1
554.9 2.01 0.882 1
584.9 1.53 0.822 1
512.3 2.69 0.937 1
585.5 1.52 0.821 1
530.3 2.41 0.917 1
594.0 1.39 0.800 1
747.6 -1.07 0.255 0
644.0 0.59 0.643 1
490.6 3.04 0.954 1
327.7 5.65 0.996 1
545.4 2.16 0.897 1
429.2 4.02 0.982 1
698.1 -0.28 0.431 1
257.8 6.77 0.999 1
427.0 4.06 0.983 1
545.9 2.16 0.896 1
464.5 3.46 0.969 1
399.9 4.49 0.989 1
442.7 3.81 0.978 1
575.4 1.68 0.843 1
613.1 1.08 0.747 1
579.8 1.61 0.834 0
808.8 -2.05 0.114 0
652.6 0.45 0.610 1
526.9 2.46 0.921 1
392.3 4.61 0.990 1
900.5 -3.52 0.029 0
582.2 1.57 0.828 0
388.8 4.67 0.991 1
512.5 2.69 0.936 1
439.9 3.85 0.979 1
447.0 3.74 0.977 1
353.4 5.23 0.995 1
502.6 2.85 0.945 1
531.3 2.39 0.916 1
333.0 5.56 0.996 1
449.8 3.69 0.976 1
490.2 3.05 0.955 1
570.4 1.76 0.854 1
494.4 2.98 0.952 1
544.7 2.17 0.898 0
330.1 5.61 0.996 1
587.0 1.50 0.817 1
588.1 1.48 0.815 1
521.7 2.54 0.927 1
678.7 0.03 0.508 1
543.4 2.20 0.900 1
523.5 2.51 0.925 1
524.1 2.51 0.924 1
548.0 2.12 0.893 1
636.5 0.71 0.669 1
550.2 2.09 0.890 1
580.2 1.61 0.833 1
324.6 5.70 0.997 1
433.4 3.96 0.981 1
695.7 -0.24 0.440 1
312.2 5.89 0.997 1
522.6 2.53 0.926 1
543.5 2.19 0.900 1
624.6 0.90 0.710 1
436.5 3.91 0.980 1
508.5 2.75 0.940 1
235.6 7.12 0.999 1
534.1 2.34 0.913 0
391.3 4.63 0.990 1
452.1 3.66 0.975 1
317.3 5.81 0.997 1
540.7 2.24 0.904 1
580.7 1.60 0.832 1
442.7 3.81 0.978 1
745.2 -1.03 0.263 1
581.2 1.59 0.831 1
620.7 0.96 0.723 1
777.0 -1.54 0.176 0
393.5 4.59 0.990 1
600.8 1.28 0.782 1
626.1 0.87 0.705 1
485.1 3.13 0.958 1
531.3 2.39 0.916 0
342.3 5.41 0.996 1
552.8 2.05 0.886 1
386.5 4.71 0.991 1
601.5 1.27 0.780 0
370.8 4.96 0.993 1
531.4 2.39 0.916 1
365.8 5.04 0.994 1
451.1 3.67 0.975 1
265.5 6.64 0.999 1
504.2 2.82 0.944 1
625.9 0.88 0.706 0
512.0 2.70 0.937 0
235.2 7.13 0.999 1
397.9 4.52 0.989 1
721.2 -0.65 0.343 1
743.0 -1.00 0.269 0
319.1 5.78 0.997 1
433.9 3.95 0.981 1
536.2 2.31 0.910 1
531.3 2.39 0.916 1
403.6 4.43 0.988 1
350.6 5.28 0.995 1
460.5 3.52 0.971 1
629.4 0.82 0.694 1
489.4 3.06 0.955 1
393.8 4.59 0.990 0
414.4 4.26 0.986 1
715.2 -0.55 0.365 0
422.5 4.13 0.984 1
594.4 1.38 0.799 1
695.0 -0.23 0.443 1
549.0 2.11 0.891 1
426.1 4.07 0.983 1
480.3 3.20 0.961 1
416.1 4.23 0.986 1
660.3 0.32 0.580 1
464.8 3.45 0.969 1
706.1 -0.41 0.399 1
406.3 4.39 0.988 1
394.1 4.58 0.990 1
506.3 2.79 0.942 1
489.4 3.06 0.955 1
414.4 4.26 0.986 1
261.4 6.71 0.999 1
490.3 3.05 0.955 1
384.2 4.74 0.991 0
639.7 0.65 0.658 1
350.7 5.28 0.995 1
646.0 0.55 0.635 1
393.8 4.59 0.990 1
527.5 2.45 0.921 1
613.6 1.07 0.745 0
412.3 4.29 0.987 1
564.7 1.85 0.865 0
473.5 3.31 0.965 1
324.8 5.69 0.997 1
447.6 3.73 0.977 1
716.0 -0.57 0.362 1
455.6 3.60 0.973 1
369.9 4.97 0.993 1
478.3 3.24 0.962 1
532.9 2.36 0.914 1
621.7 0.94 0.720 1
648.4 0.51 0.626 1
500.3 2.89 0.947 1
651.4 0.47 0.615 0
377.5 4.85 0.992 1
153.1 8.44 1.000 1
419.0 4.19 0.985 1
412.7 4.29 0.986 1
157.1 8.38 1.000 1
565.9 1.84 0.862 1
374.7 4.89 0.993 1
453.9 3.63 0.974 1
260.6 6.72 0.999 1
563.7 1.87 0.866 1
754.6 -1.18 0.235 0
361.3 5.11 0.994 1
470.0 3.37 0.967 1
762.7 -1.31 0.212 0
618.6 0.99 0.730 1
757.7 -1.23 0.226 1
485.3 3.13 0.958 1
464.4 3.46 0.970 1
413.6 4.27 0.986 1
689.7 -0.15 0.464 1
303.4 6.04 0.998 1
529.6 2.42 0.918 1
479.4 3.22 0.962 1
805.8 -2.00 0.119 0
399.7 4.50 0.989 1
271.7 6.54 0.999 1
519.0 2.59 0.930 1
480.4 3.20 0.961 1
366.8 5.02 0.993 1
504.8 2.81 0.943 1
446.9 3.74 0.977 1
389.6 4.66 0.991 1
557.1 1.98 0.878 1
438.9 3.87 0.980 0
225.4 7.28 0.999 1
109.5 9.14 1.000 1
403.6 4.43 0.988 1
692.5 -0.19 0.453 0
605.0 1.21 0.770 1
658.5 0.35 0.587 0
625.5 0.88 0.707 1
530.8 2.40 0.917 1
272.0 6.54 0.999 1
515.2 2.65 0.934 1
712.8 -0.51 0.374 0
375.0 4.89 0.993 1
642.5 0.61 0.648 0
561.1 1.91 0.871 1
598.1 1.32 0.789 0
550.9 2.08 0.888 1
562.5 1.89 0.869 1
488.1 3.08 0.956 1
355.9 5.20 0.994 1
311.8 5.90 0.997 1
491.1 3.03 0.954 1
426.2 4.07 0.983 0
358.3 5.16 0.994 1
545.3 2.17 0.897 1
507.8 2.77 0.941 0
693.8 -0.21 0.448 0
326.7 5.66 0.997 1
412.5 4.29 0.986 1
475.4 3.28 0.964 1
286.1 6.31 0.998 1
445.6 3.76 0.977 1
385.4 4.72 0.991 1
181.0 7.99 1.000 1
445.7 3.76 0.977 1
410.2 4.33 0.987 1
646.9 0.54 0.632 1
400.2 4.49 0.989 1
565.2 1.85 0.864 1
506.3 2.79 0.942 0
310.4 5.92 0.997 1
539.1 2.26 0.906 1
470.9 3.36 0.966 1
509.6 2.74 0.939 1
579.4 1.62 0.835 1
502.2 2.86 0.946 1
711.1 -0.49 0.380 0
502.4 2.85 0.945 1
307.9 5.96 0.997 1
591.3 1.43 0.807 0
312.3 5.89 0.997 1
213.5 7.47 0.999 1
371.0 4.95 0.993 1
380.3 4.81 0.992 1
399.7 4.49 0.989 1
714.2 -0.54 0.369 1
718.3 -0.60 0.354 1
589.1 1.46 0.812 1
305.1 6.01 0.998 1
489.3 3.06 0.955 1
555.2 2.01 0.882 1
466.5 3.43 0.968 1
510.4 2.72 0.938 1
351.8 5.26 0.995 1
303.2 6.04 0.998 1
685.9 -0.08 0.479 0
524.0 2.51 0.925 1
428.0 4.04 0.983 1
532.2 2.37 0.915 1
778.9 -1.57 0.172 0
537.5 2.29 0.908 1
686.4 -0.09 0.477 1
680.2 0.01 0.502 0
428.2 4.04 0.983 1
594.3 1.38 0.799 1
393.5 4.59 0.990 1
276.7 6.46 0.998 1
315.4 5.84 0.997 1
464.9 3.45 0.969 1
741.2 -0.97 0.275 1
500.9 2.88 0.947 1
372.0 4.94 0.993 1
518.7 2.59 0.930 1
585.1 1.53 0.822 1
302.1 6.06 0.998 1
336.2 5.51 0.996 1
210.3 7.53 0.999 1
305.7 6.00 0.998 1
360.0 5.13 0.994 1
444.5 3.78 0.978 1
686.2 -0.09 0.478 0
612.2 1.10 0.749 1
524.8 2.49 0.924 1
405.5 4.40 0.988 1
480.3 3.21 0.961 1
563.8 1.87 0.866 1
414.4 4.26 0.986 1
636.4 0.71 0.670 0
641.8 0.62 0.650 1
320.4 5.76 0.997 1
379.9 4.81 0.992 1
657.4 0.37 0.592 1
246.4 6.95 0.999 1
545.6 2.16 0.897 1
519.2 2.58 0.930 1
939.8 -4.15 0.016 0
366.6 5.02 0.993 1
581.7 1.58 0.830 1
723.8 -0.69 0.334 1
605.2 1.21 0.770 1
283.2 6.36 0.998 1
804.0 -1.97 0.122 0
236.5 7.11 0.999 1
328.0 5.64 0.996 1
479.9 3.21 0.961 1
802.0 -1.94 0.125 0
557.9 1.96 0.877 1
589.7 1.45 0.811 1
731.4 -0.81 0.307 0
625.8 0.88 0.706 0
312.1 5.90 0.997 1
586.8 1.50 0.818 1
470.5 3.36 0.967 1
739.1 -0.94 0.282 0
451.4 3.67 0.975 1
269.2 6.58 0.999 1
555.8 2.00 0.880 1
536.3 2.31 0.910 1
483.5 3.15 0.959 1
336.0 5.51 0.996 1
366.9 5.02 0.993 1
484.2 3.14 0.959 1
633.4 0.76 0.680 1
434.2 3.94 0.981 1
362.3 5.09 0.994 1
102.0 9.26 1.000 1
556.2 1.99 0.880 0
546.7 2.14 0.895 1
671.9 0.14 0.535 1
694.3 -0.22 0.446 0
465.7 3.44 0.969 1
670.4 0.16 0.541 0
486.2 3.11 0.957 1
389.9 4.65 0.991 1
365.6 5.04 0.994 1
436.7 3.90 0.980 1
518.3 2.60 0.931 1
482.1 3.18 0.960 1
514.5 2.66 0.934 1
514.2 2.66 0.935 1
587.5 1.49 0.816 1
447.2 3.73 0.977 1
718.2 -0.60 0.354 1
548.2 2.12 0.893 1
316.5 5.83 0.997 1
446.7 3.74 0.977 1
559.3 1.94 0.875 1
427.6 4.05 0.983 1
535.0 2.33 0.911 1
547.7 2.13 0.894 1
671.1 0.15 0.538 1
393.5 4.59 0.990 1
483.6 3.15 0.959 1
405.4 4.40 0.988 1
325.9 5.67 0.997 1
223.4 7.31 0.999 1
513.8 2.67 0.935 1
571.4 1.75 0.852 1
576.8 1.66 0.840 1
288.6 6.27 0.998 1
325.6 5.68 0.997 1
324.0 5.71 0.997 1
388.2 4.68 0.991 1
510.2 2.73 0.939 1
570.4 1.76 0.854 1
370.3 4.97 0.993 1
621.4 0.95 0.720 1
573.4 1.72 0.848 1
386.5 4.71 0.991 1
333.5 5.55 0.996 1
443.4 3.80 0.978 1
501.4 2.87 0.946 1
588.7 1.47 0.813 1
683.4 -0.04 0.489 1
365.5 5.04 0.994 1
527.3 2.45 0.921 1
675.5 0.08 0.521 0
358.0 5.16 0.994 1
454.4 3.62 0.974 1
234.5 7.14 0.999 1
526.8 2.46 0.921 1
395.0 4.57 0.990 1
354.7 5.22 0.995 1
375.2 4.89 0.993 1
549.1 2.10 0.891 1
178.9 8.03 1.000 1
425.6 4.08 0.983 1
342.1 5.42 0.996 1
349.4 5.30 0.995 1
553.4 2.04 0.884 1
535.9 2.31 0.910 1
470.8 3.36 0.966 1
562.3 1.89 0.869 1
400.5 4.48 0.989 1
198.7 7.71 1.000 1
649.9 0.49 0.620 1
453.2 3.64 0.974 1
434.2 3.94 0.981 1
508.4 2.76 0.940 1
594.8 1.37 0.798 1
546.5 2.15 0.895 0
617.7 1.01 0.732 0
492.6 3.01 0.953 1
414.4 4.26 0.986 1
443.3 3.80 0.978 1
602.9 1.24 0.776 1
365.4 5.04 0.994 1
350.6 5.28 0.995 1
530.5 2.40 0.917 1
573.0 1.72 0.848 1
245.9 6.96 0.999 1
597.4 1.33 0.791 0
487.7 3.09 0.956 0
460.3 3.53 0.971 1
643.2 0.60 0.645 0
696.1 -0.25 0.439 1
323.1 5.72 0.997 1
629.7 0.82 0.693 1
396.2 4.55 0.990 1
504.8 2.81 0.943 1
554.3 2.02 0.883 1
504.9 2.81 0.943 1
278.3 6.44 0.998 1
414.1 4.26 0.986 1
704.6 -0.38 0.405 0
510.5 2.72 0.938 1
367.3 5.01 0.993 1
392.9 4.60 0.990 1
636.2 0.71 0.671 1
535.1 2.33 0.911 1
358.6 5.15 0.994 1
486.9 3.10 0.957 1
570.7 1.76 0.853 1
730.6 -0.80 0.310 0
360.8 5.12 0.994 1
432.0 3.98 0.982 1
627.4 0.85 0.701 1
420.8 4.16 0.985 1
693.4 -0.20 0.449 0
634.1 0.74 0.678 1
250.4 6.88 0.999 1
546.2 2.15 0.896 1
284.5 6.34 0.998 1
278.0 6.44 0.998 1
496.0 2.95 0.950 1
494.1 2.99 0.952 1
481.4 3.19 0.960 1
665.5 0.24 0.560 1
527.9 2.44 0.920 1
390.4 4.64 0.990 1
503.0 2.84 0.945 1
413.0 4.28 0.986 1
476.3 3.27 0.963 1
323.0 5.72 0.997 0
469.6 3.38 0.967 1
500.6 2.88 0.947 1
386.4 4.71 0.991 1
491.9 3.02 0.953 1
393.9 4.59 0.990 1
394.6 4.58 0.990 1
457.0 3.58 0.973 1
555.7 2.00 0.881 1
456.8 3.58 0.973 1
513.1 2.68 0.936 1
623.7 0.91 0.713 1
445.5 3.76 0.977 1
471.3 3.35 0.966 1
558.6 1.95 0.876 1
307.3 5.97 0.997 1
563.2 1.88 0.867 1
410.0 4.33 0.987 1
324.4 5.70 0.997 1
234.9 7.13 0.999 1
384.5 4.74 0.991 1
319.8 5.77 0.997 1
318.7 5.79 0.997 1
717.1 -0.58 0.358 0
313.9 5.87 0.997 1
627.6 0.85 0.700 0
563.4 1.88 0.867 0
360.4 5.12 0.994 1
399.2 4.50 0.989 1
427.8 4.05 0.983 1
513.0 2.68 0.936 1
436.5 3.91 0.980 1
603.4 1.24 0.775 0
444.4 3.78 0.978 1
178.5 8.03 1.000 1
472.3 3.33 0.966 1
223.0 7.32 0.999 1
619.2 0.98 0.728 1
379.8 4.81 0.992 1
327.6 5.65 0.996 1
402.0 4.46 0.989 1
569.1 1.78 0.856 1
213.6 7.47 0.999 1
522.0 2.54 0.927 1
347.9 5.32 0.995 1
484.8 3.13 0.958 1
582.4 1.57 0.828 1
566.9 1.82 0.861 1
699.4 -0.30 0.425 0
504.3 2.82 0.944 1
386.4 4.71 0.991 1
791.8 -1.78 0.144 0
442.9 3.80 0.978 1
600.7 1.28 0.782 1
618.2 1.00 0.731 1
550.6 2.08 0.889 1
688.4 -0.12 0.469 0
456.3 3.59 0.973 1
311.1 5.91 0.997 1
434.8 3.93 0.981 1
408.3 4.36 0.987 1
532.6 2.37 0.914 1
353.5 5.23 0.995 1
582.7 1.57 0.827 0
320.7 5.76 0.997 1
735.3 -0.87 0.294 0
691.8 -0.18 0.455 0
422.3 4.13 0.984 1
505.9 2.80 0.942 1
432.9 3.96 0.981 1
314.7 5.86 0.997 1
108.1 9.16 1.000 1
482.4 3.17 0.960 1
481.1 3.19 0.961 1
719.2 -0.62 0.350 0
202.6 7.65 1.000 1
392.0 4.62 0.990 1
611.0 1.11 0.753 1
534.6 2.34 0.912 1
503.3 2.84 0.945 1
397.9 4.52 0.989 1
514.5 2.66 0.934 1
452.5 3.65 0.975 1
502.1 2.86 0.946 1
403.9 4.43 0.988 1
654.8 0.41 0.602 1
636.5 0.71 0.669 1
675.8 0.08 0.519 0
433.9 3.95 0.981 1
442.8 3.81 0.978 1
436.3 3.91 0.980 1
681.1 -0.01 0.498 0
443.4 3.79 0.978 1
499.9 2.89 0.947 1
540.1 2.25 0.904 1
582.3 1.57 0.828 1
448.0 3.72 0.976 1
819.1 -2.22 0.098 0
354.3 5.22 0.995 1
283.4 6.36 0.998 1
492.0 3.02 0.953 1
737.0 -0.90 0.289 0
493.5 2.99 0.952 1
501.6 2.86 0.946 1
608.6 1.15 0.760 1
471.9 3.34 0.966 1
500.7 2.88 0.947 1
657.6 0.37 0.591 0
499.3 2.90 0.948 1
379.2 4.82 0.992 1
542.1 2.22 0.902 1
331.0 5.59 0.996 1
278.9 6.43 0.998 1
468.6 3.39 0.967 1
329.7 5.62 0.996 1
568.5 1.79 0.857 1
465.1 3.45 0.969 1
377.2 4.86 0.992 1
716.5 -0.57 0.360 1
470.4 3.36 0.967 1
687.9 -0.12 0.471 0
469.9 3.37 0.967 1
439.2 3.86 0.979 1
539.0 2.27 0.906 1
632.2 0.77 0.684 1
870.2 -3.03 0.046 0
339.0 5.47 0.996 1
460.0 3.53 0.972 1
639.0 0.67 0.660 1
534.5 2.34 0.912 0
459.4 3.54 0.972 1
301.0 6.07 0.998 1
446.4 3.75 0.977 1
459.8 3.53 0.972 1
474.4 3.30 0.964 1
714.9 -0.55 0.366 1
420.4 4.16 0.985 1
262.2 6.69 0.999 1
556.5 1.99 0.879 1
470.1 3.37 0.967 1
457.0 3.58 0.973 1
576.3 1.67 0.842 1
407.4 4.37 0.988 1
464.2 3.46 0.970 1
572.0 1.74 0.851 1
448.9 3.71 0.976 1
343.3 5.40 0.995 1
538.9 2.27 0.906 0
488.8 3.07 0.956 1
501.7 2.86 0.946 1
650.2 0.49 0.619 1
409.0 4.35 0.987 1
308.6 5.95 0.997 1
304.2 6.02 0.998 1
340.6 5.44 0.996 1
289.0 6.27 0.998 1
479.0 3.23 0.962 1
743.2 -1.00 0.269 0
471.9 3.34 0.966 0
596.4 1.35 0.794 0
329.4 5.62 0.996 1
454.5 3.62 0.974 1
566.4 1.83 0.862 0
435.6 3.92 0.981 1
597.5 1.33 0.791 0
584.6 1.54 0.823 1
396.2 4.55 0.990 1
376.4 4.87 0.992 1
820.8 -2.24 0.096 0
458.7 3.55 0.972 1
486.9 3.10 0.957 1
478.3 3.24 0.962 1
405.3 4.40 0.988 1
354.4 5.22 0.995 1
324.7 5.69 0.997 1
615.8 1.04 0.738 0
346.0 5.35 0.995 1
580.8 1.60 0.832 1
685.7 -0.08 0.480 0
650.4 0.48 0.619 1
466.3 3.43 0.969 1
409.4 4.34 0.987 1
332.8 5.57 0.996 1
483.4 3.16 0.959 1
453.0 3.64 0.974 1
508.2 2.76 0.940 1
400.2 4.49 0.989 1
575.3 1.68 0.844 1
587.4 1.49 0.816 1
638.8 0.67 0.661 1
805.5 -2.00 0.119 1
514.1 2.66 0.935 1
404.3 4.42 0.988 1
475.0 3.29 0.964 1
254.8 6.81 0.999 1
504.4 2.82 0.944 1
425.6 4.08 0.983 1
543.7 2.19 0.899 1
447.8 3.72 0.976 1
289.9 6.25 0.998 1
491.2 3.03 0.954 1
293.8 6.19 0.998 1
613.4 1.08 0.746 1
226.2 7.27 0.999 1
465.9 3.44 0.969 1
551.0 2.07 0.888 0
270.2 6.57 0.999 1
684.7 -0.06 0.484 0
360.3 5.13 0.994 1
414.1 4.26 0.986 1
509.2 2.74 0.940 1
438.1 3.88 0.980 1
338.9 5.47 0.996 1
544.1 2.18 0.899 1
411.6 4.31 0.987 1
557.9 1.96 0.877 1
414.3 4.26 0.986 1
350.1 5.29 0.995 1
472.6 3.33 0.965 1
578.2 1.64 0.837 1
520.4 2.56 0.928 1
477.6 3.25 0.963 1
209.4 7.54 0.999 1
692.1 -0.18 0.454 0
480.3 3.20 0.961 1
363.3 5.08 0.994 1
457.4 3.57 0.973 1
523.1 2.52 0.926 0
270.4 6.56 0.999 1
587.9 1.48 0.815 1
521.1 2.55 0.928 1
656.8 0.38 0.594 1
626.0 0.87 0.706 1
409.6 4.34 0.987 1
416.4 4.23 0.986 1
344.6 5.38 0.995 1
434.6 3.94 0.981 1
620.1 0.97 0.725 1
740.5 -0.96 0.277 0
617.8 1.00 0.732 1
226.1 7.27 0.999 1
602.7 1.25 0.777 0
633.2 0.76 0.681 1
573.7 1.71 0.847 1
364.6 5.06 0.994 1
637.2 0.70 0.667 0
589.6 1.46 0.811 0
580.1 1.61 0.833 1
431.8 3.98 0.982 1
606.1 1.19 0.767 0
501.8 2.86 0.946 1
476.4 3.27 0.963 1
697.2 -0.26 0.434 1
593.5 1.39 0.801 0
425.4 4.08 0.983 1
274.8 6.49 0.998 1
569.6 1.78 0.855 1
423.2 4.12 0.984 1
614.0 1.07 0.744 1
552.4 2.05 0.886 1
489.1 3.06 0.955 1
473.9 3.31 0.965 1
434.7 3.93 0.981 1
514.3 2.66 0.935 1
328.6 5.63 0.996 1
581.5 1.59 0.830 1
430.0 4.01 0.982 1
380.4 4.80 0.992 1
439.7 3.85 0.979 1
752.2 -1.15 0.241 0
236.4 7.11 0.999 1
436.5 3.91 0.980 1
501.1 2.87 0.946 0
409.7 4.34 0.987 1
427.6 4.05 0.983 1
540.3 2.25 0.904 1
517.6 2.61 0.931 1
390.3 4.64 0.990 1
566.6 1.82 0.861 1
511.5 2.71 0.937 1
583.5 1.55 0.826 1
297.5 6.13 0.998 1
377.0 4.86 0.992 1
458.6 3.55 0.972 1
622.2 0.94 0.718 1
410.9 4.32 0.987 1
339.3 5.46 0.996 1
595.6 1.36 0.796 1
552.1 2.06 0.887 1
312.0 5.90 0.997 1
776.6 -1.54 0.177 0
698.8 -0.29 0.428 1
288.8 6.27 0.998 1
791.8 -1.78 0.144 0
240.5 7.04 0.999 1
500.5 2.88 0.947 1
323.9 5.71 0.997 1
432.5 3.97 0.981 1
436.5 3.91 0.980 1
531.8 2.38 0.915 1
326.3 5.67 0.997 1
472.4 3.33 0.965 1
380.0 4.81 0.992 1
515.9 2.64 0.933 1
365.5 5.04 0.994 1
598.5 1.31 0.788 1
490.1 3.05 0.955 1
571.7 1.74 0.851 0
344.7 5.37 0.995 1
456.0 3.59 0.973 1
534.9 2.33 0.912 1
473.3 3.32 0.965 1
720.9 -0.64 0.344 1
207.3 7.57 0.999 1
593.4 1.40 0.802 1
488.0 3.08 0.956 1
620.3 0.97 0.724 1
472.0 3.34 0.966 1
545.7 2.16 0.897 1
455.0 3.61 0.974 1
345.5 5.36 0.995 1
392.8 4.61 0.990 1
493.9 2.99 0.952 1
834.2 -2.46 0.079 0
512.7 2.69 0.936 1
346.9 5.34 0.995 1
756.3 -1.21 0.230 0
423.7 4.11 0.984 1
368.2 5.00 0.993 1
308.9 5.95 0.997 1
390.3 4.64 0.990 1
760.8 -1.28 0.217 1
614.7 1.05 0.742 1
590.6 1.44 0.809 1
506.8 2.78 0.942 1
526.2 2.47 0.922 1
569.7 1.78 0.855 1
516.7 2.62 0.932 1
508.0 2.76 0.941 1
621.4 0.95 0.720 1
462.4 3.49 0.970 1
522.2 2.53 0.927 1
358.6 5.15 0.994 1
710.9 -0.48 0.381 1
683.8 -0.05 0.487 0
228.7 7.23 0.999 1
455.3 3.61 0.974 1
564.1 1.86 0.866 0
554.5 2.02 0.883 1
594.0 1.39 0.800 0
422.7 4.13 0.984 1

Plot the observed data

What if we fit a linear model to these data

Characteristic Beta 95% CI1 p-value
(Intercept) 1.5 1.4, 1.5
I(mac_hole_thickness/100) -0.12 -0.14, -0.11
1 CI = Confidence Interval

Problem - predictions are made outside the probability domain

mac_hole_thickness prediction
0 1.47
50 1.41
100 1.35
150 1.29
200 1.22
250 1.16
300 1.10
350 1.04
400 0.98
450 0.91
500 0.85
550 0.79
600 0.73
650 0.66
700 0.60
750 0.54
800 0.48
850 0.42
900 0.35
950 0.29
1000 0.23
1050 0.17
1100 0.10
1150 0.04
1200 -0.02

Putting it all together

Test